MODEL GAME THEORY & SAPPHONS · FOLDED STARS · POLYGONS · under KEEPER ERIS

Star Polygons {n/k} · Schläfli

Join every k-th of n points on a circle and you draw the star polygon {n/k}. If gcd(n, k) = 1 it is one stroke; otherwise it falls apart into a compound, like the hexagram {6/2} = two triangles. The I-13 VM computes the gcd and every vertex with its own sine series.

Dave (t252): "i love that set up can you build some pages looking into game theory and those sappons you just built, see if you can build some folded stars and stuff"

LIT Schläfli symbol {n/k} (Poinsot 1809 studied them; the notation is Schläfli’s): n points equally spaced on a circle, each joined to the one k places on. The tip angle is 180°(1 − 2k/n): 36° for the pentagram {5/2}. k is the density, how many times the outline winds around the centre.

LIT If d = gcd(n, k) > 1 the figure is a compound of d copies of {n/d, k/d}: {6/2} = 2{3}, the Star of David; {8/2} = 2{4}, the octagram compound. For n points there are φ(n)/2 − 1 single-stroke regular stars (φ = Euler’s totient): 2 for n = 7, 1 for n = 8, 1 for n = 12.

LIT Edge length 2 sin(πk/n) on the unit circle; the inner (valley) radius of the outline is cos(πk/n)/cos(π(k − 1)/n), which is how every folded star in this batch is built.

MODEL Game tie-in: in a cyclic game where move i beats move i + k (mod n), the best-response arrows trace the star {n/k}, and the only equilibrium is to play all n moves uniformly. Rock–paper–scissors is the {3/1} case. The agents learn the {7/3} game.

MODEL Two agents, as Dave set for every world: STAR A and STAR B play the cyclic {7/3} game (each move beats the one 3 steps on, MODEL) over and over by regret matching (Hart & Mas-Colell 2000), each shifting toward the moves it regrets not having played. Their score is 1 − NashConv/2 of their empirical average play (1 = an exact Nash equilibrium), and the I-13 VM re-computes NashConv from the raw averages every time the score improves. The centrepiece is a folded star {7/3} whose fold rises with the score. Three.js r160 from a pinned CDN, devicePixelRatio capped at 2. Drag to orbit, scroll to zoom.

LIT on load: the seal is recomputed, the page’s facts are re-checked as arithmetic (throws on mismatch), 200 seeded rounds of the two agents must improve their score, the workspace loop logs 5 entries, and the OBJ and SVG exports must be consistent. Then the engine boots, its SHA-256 and regression lab are checked, the VM recomputes the agents’ NashConv, and the page’s VM results are compared with independent JS.

⏸ Pause Reseed Export training + workspace log (.json) Export star + trajectory (.svg) Export folded star (.obj)

Gallery of kept moments (WebGL snapshots):

Shared board MODEL STAR A ↔ STAR B

Background art: MODEL a generative field seeded from this page’s seal.

proving…

Seal sha256(name|slug|blurb): c0f8a1d2a26a9b65572da15c629439afb18fbf2e88cd64d9288aaddcaaf22bf7

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